Seismic imaging typically involves not only acquiring seismic data but processing the acquired seismic data. In some cases, processing requires recovering missing pieces of information from irregularly acquired seismic data. Irregularities may be caused by, for example, dead or severely corrupted seismic traces, surface obstacles, acquisition apertures, economic limits, and the like. Certain seismic processing techniques may be employed to spatially transform irregularly acquired seismic data to regularly sampled data that is easier to interpret. This regularization can involve a number of processing techniques such as interpolation and reconstruction of seismic data.
In recent years, compressive sensing theories have gained traction. One application of compressive sensing in geophysics involves seismic data reconstruction (e.g., Hennenfent and Herrmann, 2008). As an overview, compressive sensing provides conditions for when an under-determined system of equations has a desirable solution. A seismic data reconstruction problem (e.g. Stolt, 2002; Trad, 2003; Liu and Sacchi, 2004; Abma and Kabir, 2006; Ramirez et al., 2006; Naghizadeh and Sacchi, 2007; Xu et al., 2010; Kaplan et al., 2010) provides a coarse set of observed traces along with a desired set of fine spatial grid points upon which data is reconstructed. Compressive sensing theory can address such issues as 1) how many observations need to be collected, 2) where the observations should be made (i.e., sampling grid) with respect to the reconstruction grid, and 3) what mathematical dictionary (e.g., mutual coherence) should be used to represent the reconstructed data. While mutual coherence is an important metric in compressive sensing theory, it can also be expensive to compute. Descriptions and/or overviews of seismic data reconstruction can also be found in Trad, 2003; Liu and Sacchi, 2004; Abma and Kabir, 2006; Naghizadeh and Sacchi, 2007; Xu et al., 2010, the relevant parts of which are hereby incorporated by reference.
Certain data reconstruction techniques have been developed, which provide a sparse representation of reconstructed data. For example, Liu and Sacchi (2004) promote a sparse solution in wave-number domain using a penalty function constructed from inverse power spectrum of the reconstructed data. In compressive sensing, it is common to apply an l1 norm to promote some sparse representation of the reconstructed data. The l1 norm has become of particular interest due to its relation to the l0 norm which is a count of the number of non-zero elements. Theorems provide conditions for exact recovery of the reconstructed data and which, in part, rely on relationship between the l1 and l0 norms, and use of the l1 norm in a convex optimization model (Candes et al., 2006). At least one theory of compressive sensing indicates that a sparse or compressible signal can be recovered from a small number of random linear measurements by solving a convex l1 optimization problem (e.g. Baraniuk, 2007).
Compressive sensing can also provide new opportunities for survey design using an irregular sampling grid (e.g. Hennenfent and Herrmann, 2008; Kaplan et al., 2012) instead of a traditional regular grid in order to increase bandwidth and reduce cost. Generally, irregular survey design based on compressive sensing can be summarized by the following steps: 1) determine a nominal regular grid for survey area, 2) choose a subset of locations from this nominal grid in a random or randomly jittered (Hennenfent and Herrmann, 2008) fashion, 3) acquire seismic data based on chosen locations, and 4) reconstruct the data back to the original nominal grid. This approach is applicable to both shot and receiver dimensions.
In certain cases, compressive sensing using irregular acquisition grids can be used to recover significantly broader spatial bandwidth than could be obtained using a regular sampling grid. Recovered bandwidth is primarily determined according to the spacing of nominal grid for reconstruction. If a predefined nominal grid is too coarse, the reconstructed seismic data may still be aliased; if the predefined nominal grid is too fine, the time and cost savings of irregular versus regular survey design may become insignificant. In general, if there is a lack of prior information about a given survey area, then it may not be feasible to select a proper nominal grid beforehand.